On representing the positive semidefinite cone using the second-order cone
Abstract
The second-order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity. Second-order cone formulations can also be solved more efficiently than semidefinite programming in general. We consider the following question, posed by Lewis and Glineur, Parrilo, Saunderson: is it possible to express the general positive semidefinite cone using second-order cones? We provide a negative answer to this question and show that the 3x3 positive semidefinite cone does not admit any second-order cone representation. Our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3x3 positive semidefinite cone whose "second-order cone rank" grows to infinity. We also discuss the possibility of representing certain slices of the 3x3 positive semidefinite cone using the second-order cone.
Keywords
Cite
@article{arxiv.1610.04901,
title = {On representing the positive semidefinite cone using the second-order cone},
author = {Hamza Fawzi},
journal= {arXiv preprint arXiv:1610.04901},
year = {2016}
}
Comments
8 pages